I have a 2002 Alero and was told the radiator cap

Try taking some 400 grit wet dry sand paper backed with a piece of wood to the radiator filler neck sealing surface and polish in a circular motion. The replace radiator cap with a new one.

I’m happy to assist further over the phone at https://www.6ya.com/expert/kim_3202917d7c587455

There is no "radiator cap" on the Alero. It has a pressurized recovery tank (overflow bottle) integrated into the cooling system. Unlike other cooling systems where you can add coolant to the coolant reservoir when the engine is hot as there is no pressure in the reservoir, these tanks are at full system pressure and must be allowed to cool down in order to check levels and/or add coolant. This is the only place in the system that allows for coolant input.

The cap you removed on the reservoir is the radiator cap. The reservoir is part of the sealed cooling system. There is no physical cap on the radiator itself.

I’m happy to assist further over the phone at https://www.6ya.com/expert/jeremy_d728a59f986299fa

a radiator is used to cool the coolant as long as it has a top and bottom hose fitting in the right place for your engine and an oil cooler fittings for the transmission , there will be little difference in what ever else there is
some have a radiator cap and a hose to a reservoir and other do not have a cap as it is on the reservoir
the radiator shop will know the correct radiator to fit

I would add coolant to the overflow reservoir. There is not radiator cap on the radiator itself.

You shouldn't have to refill the actual radiator unless you either have a leak in it or the vacuum from it to reservoir is not good (loose hose, bad radiator cap). Adding directly to the radiator increases the chance of debris falling into the radiator and can easily clog it. Get a new radiator cap, check all around for coolant leaks and make sure hose from under cap to reservoir is tight and unclogged.

head gasket is blown

Your cooling system needs to be pressure checked to see where the leak is. certain auto parts stores rent out a pressure checker. make sure you have the cooling system full before doing the pressure test. check to see if you have a radiator cap and fill the system that way instead of using the resevoir

Your description may actually be your answer, so to speak. You mention 'generally as engine is cooling or if coolant is added too quickly'. It sounds to me like the reservoir is over-filled.
When the engine is cooling, coolant tends to flow back to the reservoir (plastic 'jug' where you add coolant). A bad radiator cap or one that was replaced with too low a PSI rating can easily allow too much to fill the reservoir and and occasionally overflow it. Most reservoirs use a rubber tube to drain excess coolant so it may appear to 'leak' near the front left wheel or left side of engine.
Replace your radiator cap. (did they pressure test the cap too?)
Make sure you check your owner's manual for the proper PSI. My sources show 15-16psi, but that sounds a little high to me. Also, make sure you have the correct (50/50) anti-freeze/water mixture.
After you replace the cap, make sure the reservoir has the right amount of coolant and isn't over-filled. (To 'fill cold' line) Drive the engine until hot, then park the truck and see if the problem is fixed. Good Luck!

You must always check the coolant hot, expansion will cause a big change in level, how low was it? top it off hot and then drive a few days and recheck the level, I don't think you have an issue here.